Integration by substitution

1. Compute each indefinite integral.

(a) $\int (1-2x)^5  dx$  solution

(b) $\displaystyle{\int \frac{x^2}{1+x^3}\, dx}$  solution

(c) $\displaystyle{\int x^2 (x^3-2)^7 \, dx}$  solution

(d) $\displaystyle{\int \frac{x}{\frac{1}{2}x^2-5}}\, dx$  solution

(e) $\int \displaystyle{ e^{-x} \, dx }$  solution

(f) $\int\displaystyle{e^{3x-1}\, dx}$   solution

(g) $\displaystyle{\int x \,\sqrt{1-x^2}   \, \,  dx}$  Solution

(h)$\displaystyle{\int (t-2)\sqrt{3t^2-12t} \, \,  dt}$   Solution

2. Find the indefinite integral.

(i) $\displaystyle{\int e^{x^5} x^4\, dx}$  solution

(ii) $\displaystyle{\int \frac{x^2-4}{x^3-12x+6} \, dx}$  solution

(iii) $\displaystyle{\int \frac{x^8}{\sqrt{x^9-7}}\, dx}$   solution

(iv) $\displaystyle{\int \frac{e^{5x}}{2+e^{5x}} \, dx}$  solution

3. Compute the following definite integral.

$\displaystyle{ \int_{-1}^1  x\,(x^2+1)^2  \, \,  dx}$  Solution 

$\displaystyle{ \int_0^1  x \, \sqrt{5x^2+4}  \, \, dx}$  Solution 

$\displaystyle{ \int_1^2  \frac{x}{(x^2+3)^2}  \, \,   dx}$   Solution 

$\displaystyle{\int_0^1 \frac{x+2}{\sqrt{x^2+4x}}  \, \,    dx}$    Solution

$\displaystyle{\int_0^4 \frac{e^{\sqrt{x}}} {\sqrt{x}}   \, \,   dx}$   Solution